THE MODULI SPACE OF CURVES AND GROMOV-WITTEN THEORY

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1 THE MODULI SPACE OF CURVES AND GROMOV-WITTEN THEORY RAVI VAKIL ABSTRACT. The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians. CONTENTS 1. Introduction 1 2. The moduli space of curves 3 3. Tautological cohomology classes on moduli spaces of curves, and their structure A blunt tool: Theorem and consequences Stable relative maps to P 1 and relative virtual localization Applications of relative virtual localization Towards Faber s intersection number conjecture 3.23 via relative virtual localization Conclusion 51 References INTRODUCTION These notes are intended to explain how Gromov-Witten theory has been useful in understanding the moduli space of complex curves. We will focus on the moduli space of smooth curves and how much of the recent progress in understanding it has come through enumerative invariants in Gromov-Witten theory, something which we take Date: Sunday, February 26, Partially supported by NSF CAREER/PECASE Grant DMS , and an Alfred P. Sloan Research Fellowship Mathematics Subject Classification: Primary 14H10, 14H81, 14N35, Secondary 14N10, 53D45, 14H15. 1

2 for granted these days, but which should really be seen as surprising. There is one sense in which it should not be surprising in many circumstances, modern arguments can be loosely interpreted as the fact that we can understand curves in general by studying branched covers of the complex projective line, as all curves can be so expressed. We will see this theme throughout the notes, from a Riemann-style parameter count in 2.2 to the tool of relative virtual localization in Gromov Witten theory in 5. These notes culminate in an approach to Faber s intersection number conjecture using relative Gromov-Witten theory (joint work with Goulden and Jackson [GJV3]). One motivation for this article is to convince the reader that our approach is natural and straightforward. We first introduce the moduli space of curves, both the moduli space of smooth curves, and the Deligne-Mumford compactification, which we will see is something forced upon us by nature, not arbitrarily imposed by man. We will then define certain geometrically natural cohomology classes on the moduli space of smooth curves (the tautological subring of the cohomology ring), and discuss Faber s foundational conjectures on this subring. We will then extend these notions to the moduli space of stable curves, and discuss Faber-type conjectures in this context. A key example is Witten s conjecture, which really preceded (and motivated) Faber s conjectures, and opened the floodgates to the last decade s flurry of developments. We will then discuss other relations in the tautological ring (both known and conjectural). We will describe Theorem (Theorem 4.1), a blunt tool for proving many statements, and Y.-P. Lee s Invariance Conjecture, which may give all relations in the tautological ring. In order to discuss the proof of Theorem, we will be finally drawn into Gromov-Witten theory, and we will quickly review the necessary background. In particular, we will need the notion of relative Gromov-Witten theory, including Jun Li s degeneration formula [Li1, Li2] and the relative virtual localization formula [GrV3]. Finally, we will use these ideas to tackle Faber s intersection number conjecture. Because the audience has a diverse background, this article is intended to be read at many different levels, with as much rigor as the reader is able to bring to it. Unless the reader has a solid knowledge of the foundations of algebraic geometry, which is most likely not the case, he or she will have to be willing to take a few notions on faith, and to ask a local expert a few questions. We will cover a lot of ground, but hopefully this article will include enough background that the reader can make explicit computations to see that he or she can actively manipulate the ideas involved. You are strongly encouraged to try these ideas out via the exercises. They are of varying difficulty, and the amount of rigor required for their solution should depend on your background. Here are some suggestions for further reading. For a gentle and quick introduction to the moduli space of curves and its tautological ring, see [V2]. For a pleasant and very detailed discussion of moduli of curves, see Harris and Morrison s foundational book [HM]. An on-line resource discussing curves and links to topology (including a glossary of important terms) is available at [GiaM]. For more on curves, Gromov-Witten theory, and localization, see [HKKPTVVZ, Chapter 22 27], which is intended for both physicists and 2

3 FIGURE 1. A complex curve, and its real cartoon mathematicians. Cox and Katz wonderful book [CK] gives an excellent mathematical approach to mirror symmetry. There is as of yet no ideal book introducing (Deligne- Mumford) stacks, but Fantechi s [Fan] and Edidin s [E] both give an excellent idea of how to think about them and work with them, and the appendix to Vistoli s paper [Vi] lays out the foundations directly, elegantly, and quickly, although this is necessarily a more serious read. Acknowledgments. I am grateful to the organizers of the June 2005 conference in Cetraro, Italy on Enumerative invariants in algebraic geometry and string theory (Kai Behrend, Barbara Fantechi, and Marco Manetti), to Fondazione C.I.M.E. (Centro Internazionale Matematico Estivo), and to the Hotel San Michele. I learned this material from my co-authors Graber, Goulden, and Jackson, and from the other experts in the field, including Carel Faber, Rahul Pandharipande, Y.-P. Lee,..., whose names are mentioned throughout this article. I thank Carel Faber, Soren Galatius, Tom Graber, Y.-P. Lee and Rahul Pandharipande for improving the manuscript. I am very grateful to Arthur Greenspoon for his many suggestions. 2. THE MODULI SPACE OF CURVES We begin with some conventions and terminology. We will work over C, although these questions remain interesting over arbitrary fields. We will work algebraically, and hence only briefly mention other important approaches to the subjects, such as the construction of the moduli space of curves as a quotient of Teichmüller space. By smooth curve, we mean a compact (also known as proper or complete), smooth (also known as nonsingular) complex curve, i.e. a Riemann surface, see Figure 1. Our curves will be connected unless we especially describe them as possibly disconnected. In general our dimensions will be algebraic or complex, which is why we refer to a Riemann surface as a curve they have algebraic/complex dimension 1. Algebraic geometers tend to draw half-dimensional cartoons of curves (see also Figure 1). The reader likely needs no motivation to be interested in Riemann surfaces. A natural question when you first hear of such objects is: what are the Riemann surfaces? How many of them are there? In other words, this question asks for a classification of curves Genus. A first invariant is the genus of the smooth curve, which can be interpreted in three ways: (i) the number of holes (topological genus; for example, the genus of the curve 3

4 in Figure 1 is 3), (ii) dimension of space of differentials (= h 0 (C, Ω C ), geometric genus), and (iii) the first cohomology group of the sheaf of algebraic functions (h 1 (C, O C ), arithmetic genus). These three notions are the same. Notions (ii) and (iii) are related by Serre duality (1) H 0 (C, F) H 1 (C, K F ) H 1 (C, K) = C where K is the canonical line bundle, which for smooth curves is the sheaf of differentials Ω C. Here F can be any finite rank vector bundle; H i refers to sheaf cohomology. Serre duality implies that h 0 (C, F) = h 1 (C, K F ), hence (taking F = K). h 0 (C, Ω C ) = h 1 (C, O C ). (We will use these important facts in the future!) As we are working purely algebraically, we will not discuss why (i) is the same as (ii) and (iii) There is a (3g 3)-dimensional family of genus g curves. Remarkably, it was already known to Riemann [R, p. 134] that there is a (3g 3)- dimensional family of genus g curves. You will notice that this can t possibly be right if g = 0, and you may know that this isn t right if g = 1, as you may have heard that elliptic curves are parametrized by the j-line, which is one-dimensional. So we will take g > 1, although there is a way to extend to g = 0 and g = 1 by making general enough definitions. (Thus there is a ( 3)-dimensional moduli space of genus 0 curves, if you define moduli space appropriately in this case as an Artin stack. But that is another story.) Let us now convince ourselves (informally) that there is a (3g 3)-dimensional family of genus g curves. This will give me a chance to introduce some useful facts that we will use later. I will use the same notation for vector bundles and their sheaves of sections. The sheaf of sections of a line bundle is called an invertible sheaf. We will use five ingredients. (1) Serre duality (1). This is a hard fact. (2) The Riemann-Roch formula. If F is any coherent sheaf (for example, a finite rank vector bundle) then h 0 (C, F) h 1 (C, F) = deg F g + 1. This is an easy fact, although I will not explain why it is true. (3) Line bundles of negative degree have no non-zero sections: if L is a line bundle of negative degree, then h 0 (C, L) = 0. Here is why: the degree of a line bundle L can be defined as follows. Let s be any non-zero meromorphic section of L. Then the degree of L is the number of zeros of s minus the number of poles of s. Thus if L has an honest non-zero section (with no poles), then the degree of s is at least 0. Exercise. If L is a degree 0 line bundle with a non-zero section s, show that L is isomorphic to the trivial bundle (the sheaf of functions) O. 4

5 (4) Hence if L is a line bundle with deg L > deg K, then h 1 (C, L) = 0 by Serre duality, from which h 0 (C, L) = deg L g + 1 by Riemann-Roch. (5) The Riemann-Hurwitz formula. Suppose C P 1 is a degree d cover of the complex projective line by a genus g curve C, with ramification r 1,..., r n at the ramification points on C. Then χ top (C) = dχ top (P 1 ) (r i 1), where χ top is the topological Euler characteristic, i.e. (2) 2 2g = 2d (r i 1). We quickly review the language of divisors and line bundles on smooth curves. A divisor is a formal linear combination of points on C, with integer coefficients, finitely many non-zero. A divisor is effective if the coefficients are non-negative. The degree of a divisor is the sum of its coefficients. Given a divisor D = n i p i (where the p i form a finite set), we obtain a line bundle O(D) by twisting the trivial bundle n i times at the point p i. This is best understood in terms of the sheaf of sections. Sections of the sheaf O(D) (over some open set) correspond to meromorphic functions that are holomorphic away from the p i ; and if n i > 0, have a pole of order at most n i at p i ; and if n i < 0, have a zero of order at least n i at p i. Each divisor yields a line bundle along with a meromorphic section (obtained by taking the function 1 in the previous sentence s description). Conversely, each line bundle with a non-zero meromorphic section yields a divisor, by taking the divisor of zeros and poles : if s is a non-zero meromorphic section, we take the divisor which is the sum of the zeros of s (with multiplicity) minus the sum of the poles of s (with multiplicity). These two constructions are inverse to each other. In short, line bundles with the additional data of a non-zero meromorphic section correspond to divisors. This identification is actually quite subtle the first few times you see it, and it is worth thinking through it carefully if you have not done so before. Similarly, line bundles with the additional data of a non-zero holomorphic section correspond to effective divisors. We now begin our dimension count. We do it in three steps. Step 1. Fix a curve C, and a degree d. Let Pic d C be the set of degree d line bundles on C. Pick a point p C. Then there is an bijection Pic 0 C Pic d C given by F F(dp). (By F(dp), we mean the twist of F at p, d times, which is the same construction sketched two paragraphs previously. In terms of sheaves, if d > 0, this means the sheaf of meromorphic sections of F, that are required to be holomorphic away from p, but may have a pole of order at most d at p. If d < 0, this means the sheaf of holomorphic sections of F that are required to have a zero of order at least d at p.) If we believe Pic d C has some nice structure, which is indeed the case, then we would expect that this would be an isomorphism. In fact, Pic d can be given the structure of a complex manifold or complex variety, and this gives an isomorphism of manifolds or varieties. Step 2: dim Pic d C = g. There are quotes around this equation because so far, Pic d C is simply a set, so this will just be a plausibility argument. By Step 1, it suffices to consider any d > deg K. Say dim Pic d C = h. We ask: how many degree d effective divisors are there 5

6 (i.e. what is the dimension of this family)? The answer is clearly d, and C d surjects onto this set (and is usually d!-to-1). But we can count effective divisors in a different way. There is an h-dimensional family of line bundles by hypothesis, and each one of these has a (d g + 1)-dimensional family of non-zero sections, each of which gives a divisor of zeros. But two sections yield the same divisor if one is a multiple of the other. Hence we get: h+(d g+1) 1 = h+d g. Thus d = h + d g, from which h = g as desired. Note that we get a bit more: if we believe that Pic d has an algebraic structure, we have a fibration (C d )/S d Pic d, where the fibers are isomorphic to P d g. In particular, Pic d is reduced (I won t define this!), and irreducible. (In fact, as many of you know, it is isomorphic to the dimension g abelian variety Pic 0 C.) Step 3. Say M g has dimension p. By fact (4) above, if d 0, and D is a divisor of degree d, then h 0 (C, O(D)) = d g + 1. If we take two general sections s, t of the line bundle O(D), we get a map to P 1 (given by p [s(p); t(p)] note that this is well-defined), and this map is degree d (the preimage of [0; 1] is precisely div s, which has d points counted with multiplicity). Conversely, any degree d cover f : C P 1 arises from two linearly independent sections of a degree d line bundle. (To get the divisor associated to one of them, consider f 1 ([0; 1]), where points are counted with multiplicities; to get the divisor associated to the other, consider f 1 ([1; 0]).) Note that (s, t) gives the same map to P 1 as (s, t ) if and only (s, t) is a scalar multiple of (s, t ). Hence the number of maps to P 1 arising from a fixed curve C and a fixed line bundle L corresponds to the choices of two sections (2(d g + 1) by fact (4)), minus 1 to forget the scalar multiple, for a total of 2d 2g + 1. If we let the the line bundle vary, the number of maps from a fixed curve is 2d 2g dim Pic d (C) = 2d g + 1. If we let the curve also vary, we see that the number of degree d genus g covers of P 1 is p + 2d g + 1. But we can also count this number using the Riemann-Hurwitz formula (2). By that formula, there will be a total of 2g + 2d 2 branch points (including multiplicity). Given the branch points (again, with multiplicity), there is a finite amount of possible monodromy data around the branch points. The Riemann Existence Theorem tells us that given any such monodromy data, we can uniquely reconstruct the cover, so we have from which p = 3g 3. p + 2d g + 1 = 2g + 2d 2, Thus there is a (3g 3)-dimensional family of genus g curves! (By showing that the space of branched covers is reduced and irreducible, we could again show that the moduli space is reduced and irreducible.) 2.3. The moduli space of smooth curves. It is time to actually define the moduli space of genus g smooth curves, denoted M g, or at least to come close to it. By moduli space of curves we mean a parameter space 6

7 for curves. As a first approximation, we mean the set of curves, but we want to endow this set with further structure (ideally that of a manifold, or even of a smooth complex variety). This structure should be given by nature, not arbitrarily defined. Certainly if there were such a space M g, we would expect a universal curve over it C g M g, so that the fiber above the point [C] representing a curve C would be that same C. Moreover, whenever we had a family of curves parametrized by some base B, say C B B (where the fiber above any point b B is some smooth genus g curve C b ), there should be a map f : B M g (at the level of sets sending b B to [C b ] M g ), and then f C g should be isomorphic to C B. We can turn this into a precise definition. The families we should consider should be nice ( fibrations in the sense of differential geometry). It turns out that the corresponding algebraic notion of nice is flat, which I will not define here. We can define M g to be the scheme such that the maps from any scheme B to it are in natural bijection with nice (flat) families of genus g curves over B. (Henceforth all families will be assumed to be nice =flat.) Some thought will convince you that only one space (up to isomorphism) exists with this property. This abstract nonsense is called Yoneda s Lemma. The argument is general, and applies to nice families of any sort of thing. Categorical translation: we are saying that this contravariant functor of families is represented by the functor Hom(, M g ). Translation: if such a space exists, then it is unique, up to unique isomorphism. If there is such a moduli space M g, we gain some additional information: cohomology classes on M g are characteristic classes for families of genus g curves. More precisely, given any family of genus g curves C B B, and any cohomology class α H (M g ), we have a cohomology class on B: if f : B M g is the moduli map, take f α. These characteristic classes behave well with respect to pullback: if C B B is a family obtained by pullback from C B B, then the cohomology class on B induced by α is the pullback of the cohomology class on B induced by α. The converse turns out to be true: any such universal cohomology class, defined for all families and well-behaved under pullback, arises from a cohomology class on M g. (The argument is actually quite tautological, and the reader is invited to think it through.) More generally, statements about the geometry of M g correspond to universal statements about all families. Here is an example of a consequence. A curve is hyperelliptic if it admits a 2-to-1 cover of P 1. In the space of smooth genus 3 curves M 3, there is a Cartier divisor of hyperelliptic curves, which means that the locus of hyperelliptic curves is locally cut out by a single equation. Hence in any family of genus 3 curves over an arbitrarily horrible base, the hyperelliptic locus are cut out by a single equation. (For scheme-theoretic experts: for any family C B B of genus 3 curves, there is then a closed subscheme of B corresponding to the hyperelliptic locus. What is an intrinsic scheme-theoretic definition of this locus?) Hence all we have to do is show that there is such a scheme M g. Sadly, there is no such scheme! We could just throw up our hands and end these notes here. There are two patches to this problem. One solution is to relax the definition of moduli space (to get the notion of coarse moduli space), which doesn t quite parametrize all families of curves. A second option is to extend the notion of space. The first choice is the more traditional one, but it is becoming increasingly clear that the second choice the better one. 7

8 This leads us to the notion of a stack, or in this case, the especially nice stack known as a Deligne-Mumford stack. This is an extension of the idea of a scheme. Defining a Deligne-Mumford stack correctly takes some time, and is rather tiring and uninspiring, but dealing with Deligne-Mumford stacks on a day-to-day basis is not so bad you just pretend it is a scheme. One might compare it to driving a car without knowing how the engine works, but really it is more like driving a car while having only the vaguest idea of what a car is. Thus I will content myself with giving you a few cautions about where your informal notion of Deligne-Mumford stack should differ with your notion of scheme. (I feel less guilty about this knowing that many analytic readers will be similarly uncomfortable with the notion of a scheme.) The main issue is that when considering cohomology rings (or the algebraic analog, Chow rings), we will take Q-coefficients in order to avoid subtle technical issues. The foundations of intersection theory for Deligne-Mumford stacks were laid by Vistoli in [Vi] (However, thanks to work of Andrew Kresch [Kr], it is possible to take integral coefficients using the Chow ring. Then we have to accept the fact that cohomology groups can be non-zero even in degree higher than the dimension of the space. This is actually something that for various reasons we want to be true, but such a discussion is not appropriate in these notes.) A smooth (or nonsingular) Deligne-Mumford stack (over C) is essentially the same thing as a complex orbifold. The main caution about saying that they are the same thing is that there are actually three different definitions of orbifold in use, and many users are convinced that their version is the only version in use, causing confusion for readers such as myself. Hence for the rest of these notes, we will take for granted that there is a moduli space of smooth curves M g (and we will make similar assumptions about other moduli spaces). Here are some facts about the moduli space of curves. The space M g has (complex) dimension 3g 3. It is smooth (as a stack), so it is an orbifold (given the appropriate definition), and we will imagine that it is a manifold. We have informally seen that it is irreducible. We make a brief brief excursion outside of algebraic geometry to show that this space has some interesting structure. In the analytic setting, M g can be expressed as the quotient of Teichmüller space (a subset of C 3g 3 homeomorphic to a ball) by a discrete group, known as the mapping class group. Hence the cohomology of the quotient M g is the group cohomology of the mapping class group. (Here it is essential that we take the quotient as an orbifold/stack.) Here is a fact suggesting that the topology of this space has some elegant structure: (3) χ(m g ) = B 2g /2g(2g 2) (due to Harer and Zagier [HZ]), where B 2g denotes the 2gth Bernoulli number. Other exciting recent work showing the attractive structure of the cohomology ring is Madsen and Weiss proof of Madsen s generalization of Mumford s conjecture [MW]. We briefly give the statement. There is a natural isomorphism between H (M g ; Q) and 8

9 1 genus 1 1 (geometric) genus 0 FIGURE 2. A pointed nodal curve, and its real cartoon H (M g+1 ; Q) for < (g 1)/2 (due to Harer and Ivanov). Hence we can define the ring we could informally denote by H (M ; Q). Mumford conjectured that this is a free polynomial ring generated by certain cohomology classes (κ-classes, to be defined in 3.1). Madsen and Weiss proved this, and a good deal more. (See [T] for an overview of the topological approach to the Mumford conjecture, and [MT] for a more technical discussion.) 2.4. Pointed nodal curves, and the moduli space of stable pointed curves. As our moduli space M g is a smooth orbifold of dimension 3g 3, it is wonderful in all ways but one: it is not compact. It would be useful to have a good compactification, one that is still smooth, and also has good geometric meaning. This leads us to extend our notion of smooth curves slightly. A node of a curve is a singularity analytically isomorphic to xy = 0 in C 2. A nodal curve is a curve (compact, connected) smooth away from a finite number of points (possibly zero), which are nodes. An example is sketched in Figure 2, in both real and cartoon form. One caution with the real picture: the two branches at the node are not tangent; this optical illusion arises from the need of our limited brains to represent the picture in three-dimensional space. A pointed nodal curve is a nodal curve with the additional data of n distinct smooth points labeled 1 through n (or n distinct labels of your choice, such as p 1 through p n ). The geometric genus of an irreducible curve is its genus once all of the nodes are unglued. For example, the components of the curve in Figure 2 have genus 1 and 0. We define the (arithmetic) genus of a pointed nodal curve informally as the genus of a smoothing of the curve, which is indicated in Figure 3. More formally, we define it as h 1 (C, O C ). This notion behaves well with respect to deformations. (More formally, it is locally constant in flat families.) Exercise (for those with enough background): If C has δ nodes, and its irreducible components have geometric genus g 1,..., g k respectively, show that k i=1 (g i 1) δ. We define the dual graph of a a pointed nodal curve as follows. It consists of vertices, edges, and half-edges. The vertices correspond to the irreducible components of the 9

10 1 FIGURE 3. By smoothing the curve of Figure 2, we see that the its genus is FIGURE 4. The dual graph to the pointed nodal curve of Figure 2 (unlabeled vertices are genus 0) curve, and are labeled with the geometric genus of the component. When the genus is 0, the label will be omitted for convenience. The edges correspond to the nodes, and join the corresponding vertices. (Note that an edge can contain a vertex to itself.) The half-edges correspond to the labeled points. The dual graph corresponding to Figure 2 is given in Figure 4. A nodal curve is said to be stable if it has finite automorphism group. This is equivalent to a combinatorial condition: (i) each genus 0 vertex of the dual graph has valence at least three, and (iii) each genus 1 vertex has valence at least one. Exercise. Prove this. You may use the fact that a genus g 2 curve has finite automorphism group, and that an elliptic curve (i.e. a 1-pointed genus 1 curve) has finite automorphism group. While you are proving this, you may as well show that the automorphism group of a stable genus 0 curve is trivial Exercise. Draw all possible stable dual graphs for g = 0 and n 5; also for g = 1 and n 2. In particular, show there are no stable dual graphs if (g, n) = (0, 0), (0, 1), (0, 2), (1, 0). Fact. There is a moduli space of stable nodal curves of genus g with n marked points, denoted M g,n. There is an open subset corresponding to smooth curves, denoted M g,n. The space M g,n is irreducible, of dimension 3g 3 + n, and smooth. (For Gromov-Witten experts: you can interpret this space as the moduli space of stable maps to a point. But this in some sense backwards, both historically, and in terms of the importance of both spaces.) 10

11 Exercise. Show that χ(m g,n ) = ( 1) n (2g+n 3)!B 2g 2g(2g 2)!, using the Harer-Zagier fact earlier (3) Strata. To each stable graph Γ of genus g with n points, we associate the subset M Γ M g,n of curves with that dual graph. This translates to the space of curves of a given topological type. Notice that if Γ is the dual graph given in Figure 4, we can obtain any curve in M Γ by taking a genus 0 curve with three marked points and gluing two of the points together, and gluing the result to a genus 1 curve with two marked points. (This is most clear in Figure 2.) Thus each M Γ is naturally the quotient of a product of M g,n s by some symmetric group. For example, if Γ is as in Figure 4, M Γ = (M 0,3 M 1,2 )/S 2. These M Γ give a stratification of M g,n, and this stratification is essentially as nice as one could hope. For example, the divisors (the closure of the codimension one strata) meet transversely along smaller strata. The dense open set M g,n is one stratum; the rest are called boundary strata. The codimension 1 strata are called boundary divisors. Notice that even if we were initially interested only in unpointed Riemann surfaces, i.e. in the moduli space M g, then this compactification forces us to consider M Γ, which in turn forces us to consider pointed nodal curves. Exercise. By computing dim M Γ, check that the codimension of the boundary stratum corresponding to a dual graph Γ is precisely the number of edges of the dual graph. (Do this first in some easy case!) 2.7. Important exercise. Convince yourself that M 0,4 = P 1. The isomorphism is given as follows. Given four distinct points p 1, p 2, p 3, p 4 on a genus 0 curve (isomorphic to P 1 ), we may take their cross-ratio λ = (p 4 p 1 )(p 2 p 3 )/(p 4 p 3 )(p 2 p 1 ), and in turn the crossratio determines the points p 1,..., p 4 up to automorphisms of P 1. The cross-ratio can take on any value in P 1 {0, 1, }. The three 0-dimensional strata correspond to these three missing points figure out which stratum corresponds to which of these three points. Exercise. Write down the strata of M 0,5, along with which stratum is in the closure of which other stratum (cf. Exercise 2.5) Natural morphisms among these moduli spaces. We next describe some natural maps between these moduli spaces. For example, given any n-pointed genus g curve (where (g, n) (0, 3), (1, 1), n > 0), we can forget the nth point, to obtain an (n 1)-pointed nodal curve of genus g. This curve may not be stable, but it can be stabilized by contracting all components that are 2-pointed genus 0 curves. This gives us a map M g,n M g,n 1, which we dub the forgetful morphism. Exercise. Create an example of a dual graph where stabilization is necessary. Also, explain why we excluded the cases (g, n) = (0, 3), (1, 1). 11

12 2.9. Important exercise. Interpret M g,n+1 M g,n as the universal curve over M g,n. (This is a bit subtle. Suppose C is a nodal curve, with node p. Which stable pointed curve with 1 marked point corresponds to p? Similarly, suppose (C, p) is a pointed curve. Which stable 2-pointed curve corresponds to p?) Given an (n 1 + 1)-pointed curve of genus g 1, and an (n 2 + 1)-pointed curve of genus g 2, we can glue the first curve to the second along the last point of each, resulting in an (n 1 + n 2 )-pointed curve of genus g 1 + g 2. This gives a map M g1,n 1 +1 M g2,n 2 +1 M g1 +g 2,n 1 +n 2. Similarly, we could take a single (n + 2)-pointed curve of genus g, and glue its last two points together to get an n-pointed curve of genus g + 1; this gives a map M g,n+2 M g+1,n. We call these last two types of maps gluing morphisms. We call the forgetful and gluing morphisms the natural morphisms between moduli spaces of curves. 3. TAUTOLOGICAL COHOMOLOGY CLASSES ON MODULI SPACES OF CURVES, AND THEIR STRUCTURE We now define some cohomology classes on these two sorts of moduli spaces of curves, M g and M g,n. Clearly by Harer and Zagier s Euler-characteristic calculation (3), we should expect some interesting classes, and it is a challenge to name some. Inside the cohomology ring, there is a subring, called the tautological (sub)ring of the cohomology ring, that consists informally of the geometrically natural classes. An equally informal definition of the tautological ring is: all the classes you can easily think of. (Of course, this isn t a mathematical statement. But we do not know of a single algebraic class in H (M g ) that can be explicitly written down, that is provably not tautological, even though we expect that they exist.) Hence we care very much about this subring. The reader may work in cohomology, or in the Chow ring (the algebraic analogue of cohomology). The tautological elements will live naturally in either, and the reader can choose what he or she is most comfortable with. In order to emphasize that one can work algebraically, and also that our dimensions and codimensions are algebraic, I will use the notation of the Chow ring A i, but most readers will prefer to interpret all statements in the cohomology ring. There is a natural map A i H 2i, and the reader should be conscious of that doubling of the index. If α is a 0-cycle on a compact orbifold X, then α is defined to be its degree. X 3.1. Tautological classes on M g, take one. A good way of producing cohomology classes on M g is to take Chern classes of some naturally defined vector bundles. 12

13 On the universal curve π : C g M g over M g, there is a natural line bundle L; on the fiber C of C g, it is the line bundle of differentials of C. Define ψ := c 1 (L), which lies in A 1 (C g ) (or H 2 (C g ) but again, we will stick to the language of A ). Then ψ i+1 A i+1 (C g ), and as π is a proper map, we can push this class forward to M g, to get the Mumford- Morita-Miller κ-class κ i := π ψ i+1, i = 0, 1,.... Another natural vector bundle is the following. Each genus g curve (i.e. each point of M g ) has a g-dimensional space of differentials ( 2.1), and the corresponding rank g vector bundle on M g is called the Hodge bundle, denoted E. (It can also be defined by E := π L.) We define the λ-classes by λ i := c i (E), i = 0,..., g. We define the tautological ring as the subring of the Chow ring generated by the κ- classes. (We will have another definition in 3.8.) This ring is denoted R (M g ) A (M g ) (or R (M g ) H 2 (M g )). It is a miraculous fact that everything else you can think of seems to lie in this subring. For example, the following generating function identity determines the λ-classes from the κ-classes in an attractive way, and incidentally serves as an advertisement for the fact that generating functions (with coefficients in the Chow ring) are a good way to package information [Fab1, p. 111]: ( λ i t i = exp i=0 i=1 B 2i κ 2i 1 2i(2i 1) t2i 1 ) Faber s conjectures. The study of the tautological ring was begun in Mumford s fundamental paper [Mu], but there was no reason to think that it was particularly well-behaved. But just over a decade ago, Carel Faber proposed a remarkable constellation of conjectures (first in print in [Fab1]), suggesting that the tautological ring has a beautiful combinatorial structure. It is reasonable to state that Faber s conjectures have motivated a great deal of the remarkable progress in understanding the topology of the moduli space of curves over the last decade. Although Faber s conjectures deal just with the moduli of smooth curves, their creation required knowledge of the compactification, and even of Gromov-Witten theory, as we will later see. A good portion of Faber s conjectures can be informally summarized as: R (M g ) behaves like the ((p, p)-part of the) cohomology ring of a (g 2)-dimensional complex projective manifold. We now describe (most of) Faber s conjectures more precisely. I have chosen to cut them into three pieces. I. Vanishing/socle conjecture. R i (M g ) = 0 for i > g 2, and R g 2 (M g ) = Q. This was proved by Looijenga [Lo] and Faber [Fab1, Thm. 2]. (Looijenga s theorem will be stated 13

14 explicitly below, see Theorem 4.5.) We will prove the vanishing part R i (M g ) = 0 for i > g 2 in 4.4, and show that R g 2 (M g ) is generated by a single element as a consequence of Theorem These statements comprise Looijenga s theorem (Theorem 4.5). The remaining part (that this generator R g 2 (M g ) is non-zero) is a theorem of Faber s, and we omit its proof. II. Perfect pairing conjecture. The analog of Poincaré duality holds: for 0 i g 2, the natural product R i (M g ) R g 2 i (M g ) R g 2 (M g ) = Q is a perfect pairing. This conjecture is currently completely open, and is only known in special cases. We call a ring satisfying I and II a Poincaré duality ring of dimension g 2. A little thought will convince you that, thanks to II, if we knew the top intersections (i.e. the products of κ-classes of total degree g 2, as a multiple of the generator of R g 2 (M g )), then we would know the complete structure of the tautological ring. Faber predicts the answer to this as well. III. Intersection number conjecture (take one). (We will give a better statement in Conjecture 3.23, in terms of a partial compactification of M g,n.) For any n-tuple of nonnegative integers (d 1,..., d n ), (4) (2g 3 + n)!(2g 1)!! (2g 1)! n j=1 (2d j + 1)!! κ g 2 = σ S n κ σ where if σ = (a 1,1 a 1,i1 )(a 2,1 a 2,i2 ) is the cycle decomposition of σ, then κ σ is defined to be j (d a j,1 + d aj,2 + +d aj,ij ). Recall that (2k 1)!! = 1 3 (2k 1) = (2k)!/2 k k!. and For example, we have κ i 1 κ g i 1 + κ g 2 = (2g 1)!! (2i 1)!!(2g 2i 1)!! κ g 2 κ g 2 1 = 1 g 1 22g 5 (g 2)! 2 κ g 2. Remarkably, Faber was able to deduce this elegant conjecture from a very limited amount of experimental data. Faber s intersection number conjecture begs an obvious question: why is this formula so combinatorial? What is the combinatorial structure behind this ring? Faber s alternate description of the intersection number conjecture (Conjecture 3.23) will be even more patently combinatorial. Faber s intersection number conjecture is now a theorem. Getzler and Pandharipande showed that it is a formal consequence of the Virasoro conjecture for the projective plane [GeP]. The Virasoro conjecture is due to the physicists Eguchi, Hori, Xiong, and also the mathematician Sheldon Katz, and deals with the Gromov-Witten invariants of some space X. (See [CK, Sect ] for a statement.) Getzler and Pandharipande show that 14

15 the Virasoro conjecture in P 2 implies a recursion among the intersection numbers on the (compact) moduli space of stable curves, which in turn is equivalent to a recursion for the top intersections in Faber s conjecture. They then show that the recursions have a unique solution, and that Faber s prediction is a solution. Givental has announced a proof of the Virasoro conjecture for projective space (and more generally Fano toric varieties) [Giv]. The details of the proof have not appeared, but Y.-P. Lee and Pandharipande are writing a book [LeeP] giving the details. This theorem is really a tour-de-force, and the most important result in Gromov-Witten theory in some time. However, it seems a roundabout and high-powered way of proving Faber s intersection number conjecture. For example, by its nature, it cannot shed light on the combinatorial structure behind the intersection numbers. For this reason, it seems worthwhile giving a more direct argument. At the end of these notes, I will outline a program for tackling this conjecture (joint with the combinatorialists I.P. Goulden and D.M. Jackson), and a proof in a large class of cases. (There are two other conjectures in this constellation worth mentioning. Faber conjectures that κ 1,..., κ [g/3] generate the tautological ring, with no relations in degrees [g/3]. Both Morita [Mo1] and Ionel [I2] have given proofs of the first part of this conjecture a few years ago. Faber also conjectures that R (M g ) satisfies the Hard Lefschetz and Hodge Positivity properties with respect to the class κ 1 [Fab1, Conj. 1(bis)]. As evidence, Faber has checked that his conjectures hold true in genus up to 21 [Fab4]. I should emphasize that this check is very difficult to do the rings in question are quite large and complicated! Faber s verification involves some clever constructions, and computer-aided computations. Morita has recently announced a conjectural form of the tautological ring, based on the representation theory of the symplectic group Sp(2g, Q) [Mo2, Conj. 1]. This is a new and explicit (and attractive) proposed description of the tautological ring. One might hope that his conjecture may imply Faber s conjecture, and may also be provable Tautological classes on M g,n. We can similarly define a tautological ring on the compact moduli space of stable pointed curves, M g,n. In fact here the definition is cleaner, and even sheds new light on the tautological ring of M g. As before, this ring includes all classes one can easily think of, and as before, it will be most cleanly described in terms of Chern classes of natural vector bundles. Before we give a formal definition, we begin by discussing some natural classes on M g,n Strata. We note first that we have some obvious (co)homology classes on M g,n, that we didn t have on M g : the fundamental classes of the (closure of the) strata. We will discuss these classes and their relations at some length before moving on. 15

16 In genus 0 (i.e., on M 0,n ), the cohomology (and Chow) ring is generated by these classes. (The reason is that each stratum of the boundary stratification is by (Zariski- )open subsets of affine space.) We will see why the tautological groups are generated by strata in Exercise 4.9. We thus have generators of the cohomology groups; it remains to find the relations. On M 0,4, the situation is especially nice. We have checked that M 0,4 is isomorphic to P 1 (Exercise 2.7), and there are three boundary points. They are homotopic (as any two points on P 1 are homotopic) and even rationally equivalent, the algebraic version of homotopic in the theory of Chow groups. By pulling back these relations by forgetful morphisms, and pushing forward by gluing morphisms, we get many other relations for various M 0,n. We dub these cross-ratio relations, although they go by many other names in the literature. Keel has shown that these are all the relations [Ke]. In genus 1, the tautological ring (although not the cohomology or Chow rings!) are again generated by strata. (We will see why in Exercise 3.28, and again in Exercise 4.9.) We again have cross-ratio relations, induced by a single (algebraic/complex) codimension 1 relation on M 0,4. Getzler proved a new (codimension 2) relation on M 1,4 [Ge1, Thm. 1.8] (now known as Getzler s relation). (It is remarkable that this relation, on an important compact smooth fourfold, parametrizing four points on elliptic curves, was discovered so late.) Via the natural morphisms, this induces relations on M 1,n for all n. Some time ago, Getzler announced that these two sorts of relations were the only relations among the strata [Ge1, par. 2]. In genus 2, there are very natural cohomology classes that are not combinations of strata, so it is now time to describe other tautological classes Other tautological classes. Once again, we can define classes as Chern classes of natural vector bundles. On M g,n, for 1 i n, we define the line bundle L i as follows. On the universal curve C g,n M g,n, the cotangent space at the fiber above [(C, p 1,..., p n )] M g,n at point p i is a one-dimensional vector space, and this vector space varies smoothly with [(C, p 1,..., p n )]. This is L i. More precisely, if s i : M g,n C g,n is the section of π corresponding to the ith marked point, then L i is the pullback by s i of the sheaf of relative differentials or the relative dualizing sheaf (it doesn t matter which, as the section meets only the smooth locus). Define ψ i = c 1 (L i ) A 1 (M g,n ). A genus g nodal curve has a g-dimensional vector space of sections of the dualizing line bundle. These vector spaces vary smoothly, yielding the Hodge bundle E g,n on M g,n. (More precisely, if π is the universal curve over M g,n, and K π is the relative dualizing line bundle on the universal curve, then E g,n := π K π.) Define λ i := c i (E g,n ) on M g,n. Clearly the restriction of the Hodge bundle and λ-classes from M g to M g are the same notions defined earlier. 16

17 Similarly, there is a more general definition of κ-classes, due to Arbarello and Cornalba [ArbC]. One might reasonably hope that these notions should behave well under the forgetful morphism π : M g,n+1 M g,n (which we can interpret as the universal curve by Exercise 2.9). Exercise. Show that there is a natural isomorphism π E g,n = Eg,n+1, and hence that π λ k = λ k. (Caution: the two λ k s in this statement are classes on two different spaces.) The behavior of the ψ-classes under pullback by the forgetful morphism has a slight twist Comparison lemma. ψ 1 = π ψ 1 + D 0,{1,n+1}. (Caution: the two ψ 1 s in the comparison lemma are classes on two different spaces!) Here D 0,{1,n+1} means the boundary divisor corresponding to reducible curves with one node, where one component is genus 0 and contains only the marked points p 1 and p n+1. The analogous statement applies with 1 replaced by any number up to n of course. Exercise (for people with more background). Prove the Comparison lemma 3.6. (Hint: First show that we have equality away from D 0,{1,n+1}. Hence ψ 1 = π ψ 1 + kd 0,{1,n+1} for some integer k, and this integer k can be computed on a single test family.) As an application: 3.7. Exercise. Show that ψ 1 on M 0,4 is O(1) (where M 0,4 = P 1, Exercise 2.7). Exercise. Express ψ 1 explicitly as a sum of boundary divisors on M 0,n. We are now ready to define the tautological ring of M g,n. We do this by defining the rings for all g and n at once Definition. The system of tautological rings (R (M g,n ) A (M g,n ) g,n ) is the smallest system of Q-algebras closed under pushforwards by the natural morphisms. This elegant definition is due to Faber and Pandharipande [FabP3, 0.1]. Define the tautological ring of any open subset of M g,n by its restriction from M g,n. In particular, we can recover our original definition of the tautological ring of M g ( 3.1). It is a surprising fact that everything else you can think of (such as ψ-classes, λ-classes and κ-classes) will lie in this ring. (It is immediate that fundamental classes of strata lie in this ring: they are pushforwards of the fundamental classes of their component spaces, cf. 2.6.) 17

18 We next give an equivalent description of the tautological groups, which will be convenient for many of our arguments, because we do not need to make use of the multiplicative structure. In this description, the ψ-classes play a central role Definition [GrV3, Defn. 4.2]. The system of tautological rings (R (M g,n ) A (M g,n ) g,n ) is the smallest system of Q-vector spaces closed under pushforwards by the natural morphisms, such that all monomials in ψ 1,..., ψ n lie in R (M g,n ). The equivalence of Definition 3.8 and Definition 3.9 is not difficult (see for example [GrV3]) Faber-type conjectures for M g,n, and the conjecture of Hain-Looijenga-Faber- Pandharipande. In analogy with Faber s conjecture, we have the following Conjecture. R (M g,n ) is a Poincaré-duality ring of dimension 3g 3 + n. This was first asked as a question by Hain and Looijenga [HLo, Question 5.5], first stated as a speculation by Faber and Pandharipande [FabP1, Speculation 3] (in the case n = 0), and first stated as a conjecture by Pandharipande [P, Conjecture 1]. In analogy with Faber s conjecture, we break this into two parts. I. Socle conjecture. R 3g 3+n (M g,n ) = Q. This is obvious if we define the tautological ring in terms of cohomology: H 2(3g 3+n) (M g,n ) = Q, and the zero-dimensional strata show that the tautological zero-cycles are not all zero. However, in the tautological Chow ring, the socle conjecture is not at all obvious. Moreover, the conjecture is not true in the full Chow ring A 0 (M 1,11 ) is uncountably generated, while the conjecture states that R 0 (M 1,11 ) has a single generator. (By R 0, we of course mean R 3g 3+n.) We will prove the vanishing conjecture in 4.6. II. Perfect pairing conjecture For 0 i 3g 3 + n, the natural product R i (M g,n ) R 3g 3+n i (M g,n ) R 3g 3+n (M g,n ) = Q is a perfect pairing. (We currently have no idea why this should be true.) Hence, in analogy with Faber s conjecture, if this conjecture were true, then we could recover the entire ring by knowing the top intersections. This begs the question of how to compute all top intersections Fact/recipe (Mumford and Faber). If we knew the top intersections of ψ-classes, we would know all top intersections. In other words, there is an algorithm to compute all top intersections if we knew the numbers (5) ψ a 1 1 ψan n, ai = 3g 3 + n. M g,n 18

19 (This is a worthwhile exercise for people with some familiarity with the moduli space of curves.) This is the basis of Faber s wonderful computer program [Fab2] computing top intersections of various tautological classes. For more information, see [Fab3]. This construction is useful in understanding the definition (Defn. 3.9) of the tautological group in terms of the ψ-classes. Until a key insight of Witten s, there was no a priori reason to expect that these numbers should behave nicely. We will survey three methods of computing these numbers: (i) partial results in low genus; (ii) Witten s conjecture; and (iii) via the ELSV formula. A fourth (attractive) method was given in Kevin Costello s thesis [C] Top intersections on M g,n : partial results in low genus. Here are two crucial relations among top intersections. Dilaton equation. If M g,n exists (i.e. there are stable n-pointed genus g curves, or equivalently 2g 2 + n > 0), then ψ β 1 1 ψβ 2 2 ψβn n ψ n+1 = (2g 2 + n) ψ β 1 1 ψβn n. M g,n+1 M g,n String equation. If 2g 2 + n > 0, then ψ β 1 1 ψβ 2 2 ψβn n = n M g,n+1 i=1 M g,n ψ β 1 1 ψβ 2 (where you ignore terms where you see negative exponents). 2 ψβ i 1 i ψ βn n Exercise (for those with more experience). Prove these using the Comparison lemma 3.6. Equipped with the string equation alone, we can compute all top intersections in genus 0, i.e. M 0,n ψ β 1 1 ψβn n where β i = n 3. (In any such expression, some β i must be 0, so the string equation may be used.) Thus we can recursively solve for these numbers, starting from the base case M 0,3 ψ 0 1 ψ0 2 ψ0 3 = 1. Exercise. Show that ( ) ψ a 1 n 3 1 ψan n =. M 0,n a 1,, a n In genus 1, the story is similar. In this case, we need both the string and dilaton equation. Exercise. Show that any integral ψ β 1 1 ψβn n M 1,n can be computed using the string and dilaton equation from the base case M 1,1 ψ 1 = 1/24. 19

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